Advertisement

Polyconvex Energies for Trigonal, Tetragonal and Cubic Symmetry Groups

  • Jörg Schröder
  • Patrizio Neff
  • Vera Ebbing
Conference paper
Part of the IUTAM Bookseries book series (IUTAMBOOK, volume 21)

Abstract

In large strain elasticity the existence of minimizers is guaranteed if the variational functional to be minimized is sequentially weakly lower semicontinuous (s.w.l.s.) and coercive. Therefore, for the description of hyperelastic materials polyconvex functions which are always s.w.l.s. should be preferably used. A variety of isotropic and anisotropic polyconvex energies, in particular for the triclinic, monoclinic, rhombic and transversely isotropic symmetry groups, already exist. In this contribution we propose a new approach for the description of trigonal, tetragonal and cubic hyperelastic materials in the framework of polyconvexity. The anisotropy of the material is described by invariants in terms of the right Cauchy’Green tensor and a specific fourth-order structural tensor. In order to show the adaptability of the introduced polyconvex energies for the approximation of real anisotropic material behavior we focus on the fitting of a trigonal fourth-order tangent moduli near the reference state to experimental data.

Keywords

Lithium Niobate Structural Tensor Hyperelastic Material Tensor Function Green Tensor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ball, J.M.: Convexity conditions and existence theorems in non-linear elasticity. Archive for Rational Mechanics and Analysis63, 1977, 337–403.zbMATHCrossRefGoogle Scholar
  2. 2.
    Betten, J.: Integrity basis for a second-order and a fourth-order tensor. International Journal of Mathematics and Mathematical Sciences5(1), 1982, 87–96.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Betten, J.: Recent advances in applications of tensor functions in solid mechanics. Advances in Mechanics14(1), 1991, 79–109.MathSciNetGoogle Scholar
  4. 4.
    Betten, J.: Anwendungen von Tensorfunktionen in der Kontinuumsmechanik anisotroper Materialien. Zeitschrift für Angewandte Mathematik und Mechanik78(8), 1998, 507–521.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Betten, J. and Helisch, W.: Irreduzible Invarianten eines Tensors vierter Stufe. Zeitschrift für Angewandte Mathematik und Mechanik72(1), 1992, 45–57.zbMATHMathSciNetGoogle Scholar
  6. 6.
    Betten, J. and Helisch, W.: Tensorgeneratoren bei Systemen von Tensoren zweiter und vierter Stufe. Zeitschrift für Angewandte Mathematik und Mechanik76(2), 1996, 87–92.zbMATHMathSciNetGoogle Scholar
  7. 7.
    Boehler, J.P.: Lois de comportement anisotrope des milieux continus. Journal de Mécanique17(2), 1978, 153–190.zbMATHMathSciNetGoogle Scholar
  8. 8.
    Boehler, J.P.: A simple derivation of representations for non-polynomial constitutive equations in some cases of anisotropy. Zeitschrift für Angewandte Mathematik und Mechanik59, 1979, 157–167.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Boehler, J.P.: Introduction to the invariant formulation of anisotropic constitutive equations. In: J.P. Boehler (Ed.), Applications of Tensor Functions in Solid Mechanics, CISM Courses and Lectures, Vol. 292, Springer, 1987, pp. 13–30.Google Scholar
  10. 10.
    Ebbing, V., Schröder, J. and Neff, P.: On the construction of anisotropic polyconex energy densities. In: Proceedings in Applied Mathematics and Mechanics, Vol. 7, 2007, pp. 4060,009–4060,010.CrossRefGoogle Scholar
  11. 11.
    Ebbing, V., Schröder, J. and Neff, P.: Polyconvex models for arbitrary anisotropic materials. In: Proceedings in Applied Mathematics and Mechanics, Vol. 8, 2008, pp. 10,415–10,416.CrossRefGoogle Scholar
  12. 12.
    Ebbing, V., Schröder, J. and Neff, P.: Approximation of anisotropic elasticity tensors at the reference state with polyconvex energies. Archive of Applied Mechanics79, 2009, 651–657.CrossRefGoogle Scholar
  13. 13.
    Jarić, J.P., Kuzmanović, D. and Golubović, Z.: On tensors of elasticity. Theoretical and Applied Mechanics35(1–3), 2008, 119–136.zbMATHMathSciNetGoogle Scholar
  14. 14.
    Kambouchev, N., Fernandez, J. and Radovitzky, R.: A polyconvex model for materials with cubic symmetry. Modelling and Simulation in Material Science and Engineering15, 2007, 451–467.CrossRefGoogle Scholar
  15. 15.
    Liu, I.S.: On representations of anisotropic invariants. International Journal of Engineering Science20, 1982, 1099–1109.zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Neumann, F.E.: Vorlesungen über die Theorie der Elastizität der festen Körper und des Lichtäthers, Teubner, 1885.Google Scholar
  17. 17.
    Schröder, J. and Neff, P.: On the construction of polyconvex anisotropic free energy functions. In: C. Miehe (Ed.), Proceedings of the IUTAM Symposium on Computational Mechanics of Solid Materials at Large Strains, Kluwer Academic Publishers, 2001, pp. 171–180.Google Scholar
  18. 18.
    Schröder, J. and Neff, P.: Invariant formulation of hyperelastic transverse isotropy based on polyconvex free energy functions. International Journal of Solids and Structures40, 2003, 401–445.zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Schröder, J., Neff, P. and Ebbing, V.: Anisotropic polyconvex energies on the basis of crystallographic motivated structural tensors. Journal of the Mechanics and Physics of Solids56(12), 2008, 3486–3506.zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Schwefel, H.P.: Evolution and Optimum Seeking, Wiley, 1996.Google Scholar
  21. 21.
    Simmons, G. and Wang, H.: Single Crystal Elastic Constants and Calculated Aggregate Properties, The MIT Press, Massachusetts Institute of Technology, 1971.Google Scholar
  22. 22.
    Smith, G.F.: On a fundamental error in two papers of C.-C. Wang “On representations for isotropic functions, Parts I and II”. Archive for Rational Mechanics and Analysis36, 1970, 161–165.zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Smith, G.F.: On isotropic functions of symmetric tensors, skew-symmetric tensors and vectors. International Journal of Engineering Science9, 1971, 899–916.zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Spencer, A.J.M.: Theory of invariants. In: A. Eringen (Ed.), Continuum Physics, Vol. 1, Academic Press, 1971, pp. 239–353.Google Scholar
  25. 25.
    Wang, C.C.: On representations for isotropic functions. Part I. Isotropic functions of symmetric tensors and vectors. Archive for Rational Mechanics and Analysis33, 1969, 249–267.CrossRefMathSciNetGoogle Scholar
  26. 26.
    Wang, C.C.: On representations for isotropic functions. Part II. Isotropic functions of skew-symmetric tensors, symmetric tensors, and vectors. Archive for Rational Mechanics and Analysis33, 1969, 268–287.CrossRefMathSciNetGoogle Scholar
  27. 27.
    Wang, C.C.: A new representation theorem for isotropic functions: An answer to Professor G.F. Smith’s criticism of my papers on representations for isotropic functions. Part 1. Scalar-valued isotropic functions. Archive for Rational Mechanics and Analysis36, 1970, 166–197.zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Wang, C.C.: A new representation theorem for isotropic functions: An answer to professor G.F. Smith’s criticism of my papers on representations for isotropic functions. Part 2. Vector-valued isotropic functions, symmetric tensor-valued isotropic functions, and skew-symmetric tensor-valued isotropic functions. Archive for Rational Mechanics and Analysis36, 1970, 198–223.zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Wang, C.C.: Corrigendum to my recent papers on “Representations for isotropic functions”. Archive for Rational Mechanics and Analysis43, 1971, 392–395.CrossRefMathSciNetGoogle Scholar
  30. 30.
    Xiao, H.: On isotropic extension of anisotropic tensor functions. Zeitschrift für Angewandte Mathematik und Mechanik76(4), 1996, 205–214.zbMATHGoogle Scholar
  31. 31.
    Zhang, J. and Rychlewski, J.: Structural tensors for anisotropic solids. Archives of Mechanics42, 1990, 267–277.zbMATHMathSciNetGoogle Scholar
  32. 32.
    Zheng, Q.S.: Theory of representations for tensor functions – A unified invariant approach to constitutive equations. Applied Mechanics Reviews47, 1994, 545–587.CrossRefGoogle Scholar
  33. 33.
    Zheng, Q.S. and Spencer, A.J.M.: Tensors which characterize anisotropies. International Journal of Engineering Science31(5), 1993, 679–693.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Institut für Mechanik, Abteilung Bauwissenschaften, Fakultät für IngenieurwissenschaftenUniversität Duisburg-EssenEssenGermany
  2. 2.Lehrstuhl für Nichtlineare Analysis und Modellierung, Fakultät für MathematikUniversität Duisburg-EssenEssenGermany

Personalised recommendations